Advanced Calculus fi..

422 Advanced Calculus, Fifth EditionIn other words,-% t . 7"-If xz, un(x) and xzl vn(x) are uniformly convergent forTHEOREM 34a 5 x 5 b and h(x) is continuous for a 5 x 5 b, then the seriesare uniformly convergent for a 5 x 5 b.Proof. Let f (x) and g(x) be the sums of C un(x) and C vn(x),respectively; letSn(x) and Qn(x) be the corresponding partial sums. The nth partial sum of C(un +v,)is Sn + Qn. Let N be chosen, for given 6, so thatforn 2 N anda 5 x 5 b.Then1 1ISn(x) - f (x)l < 26' IQn(x) - g(x)l < I€,Thus C(un + u,) converges uniformly to f (x) + g(x). A similar proof applies tothe difference.Since h(x) is continuous for a 5 x 5 b, it is necessarily bounded: Ih(x)l 5 Mfor a 5 x 5 b. HenceI'Ih(x)Sn(x) - h(x)f (x)l = Ih(x)llSn(x) - f (x)l < M . 26 < Mefor n 2 N as before. This shows that the series C h(x)un converges uniformlyto h(x) f (x). It should be noted that actually only the boundedness of h(x) wasrequired.?By a power series in powers of x is meant a series of the formwhere co, cl, . . . , c,, . . . are constants. By a power series in powers of (x - a) ismeant a series:00Ec,(x - a)n = co +cl(x -a)+ ... +cn(x -a)" +n=O. . a . (6.36)